What type of company do I call for buried gutter drain pipe repair?

[The Composer]

We had a very rainy weekend, in fact, on Saturday alone we received about 5 inches of rain. As a consequence, some water seeped into my basement and wet some carpet.

To make a long story short, I believe my underground gutter drain pipes are clogged (after inspection on Sunday) and water backed up and poured down the foundation. What type of company do I call for the gutter drain line repair? Does Roto-Rooter do that kind of work? I've only thought of them as indoor plumbing or sewer work.

The buried pipe is the corrugated black pipe. I thought about renting a snake, but am unsure whether or not the snake would damage the pipe.

Any suggestions?